Advancements in disciplines ranging from atomic physics to various branches of condensed matter physics are being employed to fabricate a variety of different diamond-based materials for use in many different technological applications. Diamond has a crystal lattice structure comprising two interpenetrating face-centered cubic lattices of carbon atoms. FIG. 1A shows a unit cell 100 of a diamond-crystal lattice. In FIG. 1A, each carbon atom, represented by a sphere, is covalently bonded to four adjacent carbon atoms, each covalent bond is represented by a rod connecting two spheres. As shown in FIG. 1A, a carbon atom 102 is covalently bonded to four carbon atoms 103-106. In general, diamond has a number of potentially useful properties. For example, diamond is transparent from the ultraviolet to the far infrared of the electromagnetic spectrum and has a relatively high refractive index of about 2.42. Diamond may also be a suitable replacement for silicon in silicon-based semiconductor devices. For example, silicon has an electronic bandgap of about 1.12 eV and starts to show signs of thermal stress at about 100° C., while diamond has a larger electronic bandgap ranging from about 5 eV to about 7 eV and a high Debye temperature ranging from about 1550° C. to about 1930° C.
Certain atomic-vacancy systems, called “color centers,” embedded in diamond may have potential applications in quantum computing and quantum information processing. For example, a nitrogen-vacancy (“NV”) center embedded in diamond is a type of color center that may be used to store a quantum bit of information. FIG. 1B shows an NV center embedded in a diamond-crystal lattice 110. The NV-center comprises a nitrogen atom 112 next to a carbon vacancy 114. The nitrogen atom 112 is covalently bonded to three carbon atoms 116-118. The nitrogen atom 112 can also be located within the diamond-crystal lattice 110 at the positions occupied by carbons 116-118. NV centers can be created in a nitrogen rich diamond by irradiation and subsequent annealing at temperatures above 550° C. The radiation creates vacancies in the diamond and subsequent annealing causes the vacancies to migrate towards nitrogen atoms to produce an NV center. Alternatively, NV centers can be created in diamond using N+ ion implantation.
When an electromagnetic field interacts with an NV center, there is a periodic exchange, or oscillation, of energy between the electromagnetic field and the electronic energy levels of the NV center. Such oscillations, which are called “Rabi oscillations,” are associated with oscillations of the NV center electronic energy level populations and quantum-mechanical probability amplitudes of the NV center electronic energy states. Rabi oscillations can be interpreted as an oscillation between absorption and stimulated emission of photons. The Rabi frequency, denoted by Ω, represents the number of times these oscillations occur per unit time (multiplied by the quantity 2π).
FIG. 1C illustrates an energy-level diagram of electronic states of a negatively charged NV center. The energy-level diagram has a Λ-shaped configuration comprising three ground states coupled to a common excited state. The three ground 3A2 states comprise a first ground state |1 with a lowest energy level 122, and a pair of nearly degenerate ground states |2 and |3 with energy levels 124 and 126, respectively. The excited 3E state |4 has a number of different energy levels 128. Note that the exact structure of the 3E state depends on the strain or mechanical effects in the diamond crystal. A parameter δ1 is the laser frequency detuning for a |1→|4 transition, a parameter δ2 is the laser frequency detuning for a |2→|4 transition, a parameter δ23 is the |2−|3 energy splitting, and Ωi represent Rabi frequencies, which are proportional to the square root of the laser intensities. For a description of experimental investigations of NV centers, see “The nitrogen-vacancy center in diamond re-visted,” by N. B. Manson et al., preprint: http://arxiv.org/abs/cond-mat/0601360; “Coherent population trapping with a single spin in diamond,” by Charles Santori et al., preprint: http://arxiv.org/abs/quant-ph/0607147; and “Coherent population trapping in Diamond N-V centers at zero magnetic field,” by Charles Santori et al., preprint: http://arxiv.org/abs/cond-mat/0602573.
When spontaneous emission is taken into consideration, the state of the NV center is, in general, represented by a linear superposition of states:|ψ=a1|1+a2|2+a3|3where a1, a2, and a3 are probability amplitudes. In other words, only linear superpositions of states with zero probability amplitude (a4=0) in the excited state |4 are stable. These stable linear superposition of states are also called “dark states.” When δ1 equals δ2, the state of the NV center is in a dark state represented by:
                          ψ        1            〉        =                                                                                        Ω                  2                                ⁢                                                    1                  〉                                            -                              Ω                1                                                          ⁢          2                〉                                          Ω            1            2                    +                      Ω            2            2                                ,Beginning with the NV center in the state |1, the Rabi frequency Ω2 is finite and the Rabi frequency Ω1 is zero. However, turning off the electromagnetic field associated with the Rabi frequency Ω2 while turning on the electromagnetic field associated with the Rabi frequency Ω1 places the NV center in the state |2. When δ1 equals δ2+δ23, the state of the NV center is in a dark state represented by:
                ψ      2        〉    =                                                                        Ω                3                            ⁢                                              1                〉                                      -                          Ω              1                                                ⁢        3            〉                                Ω          1          2                +                  Ω          3          2                    Likewise, beginning with the NV center in the state |1, Ω3 is finite and Ω1 is zero. Turning off the electromagnetic field associated with the Rabi frequency Ω3 while turning on the electromagnetic field associated with the Rabi frequency Ω1 places the NV center in the state |3. For the degenerate case of δ23 equal to 0, Ω2 equals Ω3 and the state of the NV center can be in a dark state represented by either |ψ1 or |ψ2. The dark states |ψ1 and |ψ2 have a number of potential applications in quantum information processing because either dark state of an NV center can be used to store a quantum bit of information.
The NV centers are appealing for quantum information processing because the NV center has a relatively long-lived spin coherence time and a possibility of large-scale integration into semiconductor processing technology. For example, an NV center electron spin coherence time of 58 μs has been observed at room temperature. See “Long coherence times at 300K for nitrogen-vacancy center spins in diamond grown by chemical vapor deposition,” by A. Kennedy et al., App. Phys. Lett. 83, 4190-4192 (2003). NV centers may have relatively long-lived spin coherence because the lattice comprises primarily 12C, which has zero nuclear spin. In addition, a single photon can be generated from an NV center at room temperature, which has established NV centers as potential photon sources in quantum computing and quantum communications. See “Stable solid-state source of single photons,” by C. Kurtsiefer et al., Phys. Rev. Lett. 85, 290-293 (2000) and “Room temperature stable single photon source,” by A. Beveratos et al., Eur. Phys. J. D 18, 191-196 (2002). Physicists, computer scientists, and engineers have, therefore, recognized a need for color-center-based quantum computing systems that can be used in a variety of quantum-based applications, such as quantum computing, quantum information processing, and quantum communications.